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• - [Instructor] You've likely heard the concept

• of even and odd numbers, and what we're going to do

• in this video is think about even and odd functions.

• And as you can see or as you will see,

• there's a little bit of a parallel between the two,

• but there's also some differences.

• So let's first think about what an even function is.

• One way to think about an even function is that

• if you were to flip it over the y-axis,

• that the function looks the same.

• So here's a classic example of an even function.

• It would be this right over here,

• is on the y-axis.

• This is an even function.

• So this one is maybe the graph

• of f of x is equal to x squared.

• And notice, if you were to flip it over the y-axis,

• you're going to get the exact same graph.

• Now, a way that we can talk about that mathematically,

• when we introduced the idea of reflection,

• to say that a function is equal

• to its reflection over the y-axis,

• that's just saying that f of x is equal to f of negative x.

• Because if you were to replace your x's with a negative x,

• that flips your function over the y-axis.

• Now, what about odd functions?

• So odd functions, you get the same function

• if you flip over the y- and the x-axes.

• So let me draw a classic example of an odd function.

• Our classic example would be

• f of x is equal to x to the third,

• is equal to x to the third,

• and it looks something like this.

• So notice, if you were to flip first over the y-axis,

• you would get something that looks like this.

• So I'll do it as a dotted line.

• If you were to flip just over the y-axis,

• it would look like this.

• And then if you were to flip that over the x-axis,

• well, then you're going to get the same function again.

• Now, how would we write this down mathematically?

• Well, that means that our function is equivalent to

• not only flipping it over the y-axis,

• which would be f of negative x,

• but then flipping that over the x-axis,

• which is just taking the negative of that.

• So this is doing two flips.

• So some of you might be noticing a pattern

• or think you might be on the verge of seeing a pattern

• that connects the words even and odd with the notions

• that we know from earlier in our mathematical lives.

• I've just shown you an even function

• where the exponent is an even number,

• and I've just showed you an odd function

• where the exponent is an odd number.

• Now, I encourage you to try out many, many more polynomials

• and try out the exponents,

• but it turns out that if you just have f of x is equal to,

• if you just have f of x is equal to x to the n,

• then this is going to be an even function if n is even,

• and it's going to an odd function if n is odd.

• So that's one connection.

• Now, some of you are thinking,

• "Wait, but there seem to be a lot of functions

• "that are neither even nor odd."

• And that is indeed the case.

• For example, if you just had the graph x squared plus two,

• this right over here is still going to be even.

• 'Cause if you flip it over,

• you have the symmetry around the y-axis.

• You're going to get back to itself.

• But if you had x minus two squared,

• which looks like this,

• x minus two, that would shift two to the right,

• it'll look like that.

• That is no longer even.

• Because notice, if you flip it over the y-axis,

• you're no longer getting the same function.

• So it's not just the exponent.

• It also matters on the structure of the expression itself.

• If you have something very simple, like just x to the n,

• well, then that could be or that would be even or odd

• depending on what your n is.

• Similarly, if we were to shift this f of x,

• if we were to even shift it up, it's no longer,

• it is no longer, so if this is x to the third,

• let's say, plus three,

• this is no longer odd.

• Because you flip it over once, you get right over there.

• But then you flip it again, you're going to get this.

• You're going to get something like this.

• So you're no longer back to your original function.

• Now, an interesting thing to think about,

• can you imagine a function that is both even and odd?

• So I encourage you to pause that video,

• or pause the video and try to think about it.

• Is there a function where f of x is equal to f of negative x

• and f of x is equal to the negative of f of negative x?

• Well, I'll give you a hint,

• or actually I'll just give you the answer.

• Imagine if f of x is just equal to the constant zero.

• Notice, this thing is just a horizontal line,

• just like that, at y is equal to zero.

• And if you flip it over the y-axis,

• you get back to where it was before.

• Then if you flip it over the x-axis,

• again, then you're still back to where you were before.

• So this over here is both even and odd,

• a very interesting case.

- [Instructor] You've likely heard the concept

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# 函數對稱性介紹｜函數的變換｜代數2｜可汗學院 (Function symmetry introduction | Transformations of functions | Algebra 2 | Khan Academy)

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林宜悉 發佈於 2021 年 01 月 14 日